Past Stochastic Analysis Seminar

We follow Arnold's approach of Euler equation as a geodesic on the group of

diffeomorphisms. We construct a geometrical Brownian motion on this group in the

case of the two dimensional torus, and prove the global existence of a

stochastic perturbation of Euler equation (joint work with F. Flandoli and P.

Malliavin).

Other diffusions allow us to obtain the deterministic Navier-Stokes equation

as a solution of a variational problem (joint work with F. Cipriano).

In St Anne's College

In this talk, the convergence analysis of a class of weak approximations of

solutions of stochastic differential equations is presented. This class includes

recent approximations such as Kusuoka's moment similar families method and the

Lyons-Victoir cubature on Wiener Space approach. It will be shown that the rate

of convergence depends intrinsically on the smoothness of the chosen test

function. For smooth functions (the required degree of smoothness depends on the

order of the approximation), an equidistant partition of the time interval on

which the approximation is sought is optimal. For functions that are less smooth

(for example Lipschitz functions), the rate of convergence decays and the

optimal partition is no longer equidistant. An asymptotic rate of convergence

will also be presented for the Lyons-Victoir method. The analysis rests upon

Kusuoka-Stroock's results on the smoothness of the distribution of the solution

of a stochastic differential equation. Finally, the results will be applied to

the numerical solution of the filtering problem.

We consider multi-dimensional Gaussian processes and give a novel, simple and

sharp condition on its covariance (finiteness of its two dimensional rho-variation,

for some rho <2) for the existence of "natural" Levy areas and higher iterated

integrals, and subsequently the existence of Gaussian rough paths. We prove a

variety of (weak and strong) approximation results, large deviations, and

support description.

Rough path theory then gives a theory of differential equations driven by

Gaussian signals with a variety of novel continuity properties, large deviation

estimates and support descriptions generalizing classical results of

Freidlin-Wentzell and Stroock-Varadhan respectively.

(Joint work with Nicolas Victoir.)

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